Seven Factorial

“I amsick will not to choir because i have a heache. i Hope its very and i am so sorry

love,

blue”

IT WAS ON THE DESKTOP. THIS IS WHAT I SENT.

I crocheted a series of hyperbolic planes and strung them up into a garland. Each module is one more round than the one before, and each round has twice the stitches as the round before.

I made eight iterations; the largest one took an entire skein of cotton yarn. I really liked how this shows the change in curvature as cross section of the plane expands.

Akatsuki's IR2 camera acquired this view of the night side of Venus. Infrared energy from hot, mid-altitude clouds shows up bright, while higher clouds that block the heat appear dark. The dark sawtooth at center appears to be a turbulent boundary. The planet's sunlit crescent is overexposed at upper right.

Most famous constants announce themselves immediately. For example, π appears wherever circles show up, e emerges from growth and calculus, and so on. But there's a stranger hiding deep within this infinite series:

ζ(3) = 1 + 1 / 2^3 + 1 / 3^3 + 1 / 4^3 + . . .

The number this sum converges to is called Apéry's constant, and despite looking innocent, it resisted proof for centuries. Mathematicians strongly suspected that it was irrational but nobody could prove it until 1978.

The Zeta Function Apéry's constant comes from one of the most important objects in mathematics: the Riemann zeta function.

For real numbers s > 1,

ζ(s) = ∑_{n=1}^∞ 1 / n^s

(the infinite series of 1 / 1 ^ s + 1 / 2 ^ s + 1 / 3 ^ s + . . . )

At first glance, this is just an infinite sum. But the zeta function secretly connects prime numbers, complex analysis, quantum physics, probability, cryptography, and the distribution of the primes themselves.

Some values are beautifully understood. For example,

ζ(2) = π^2 / 6

ζ(4) = π^4 / 90

In fact, every even positive integer produces a formula involving powers of π. But the odd inputs are another story.

Nobody knows a comparably elegant formula for

ζ(3), ζ(5), ζ(7), . . .

These numbers are mysterious, and ζ(3) became the first "battleground".

Roger Apéry's Bombshell

Numerically, Apéry's constant equals approximately

1.202056903159694...

The question sounds deceptively simple: Is this number rational? For over 200 years, nobody knew. That's remarkable because Euler had solved the analogous problem for ζ(2) in the 1700s.

In 1978, French Mathematician Roger Apéry announced that ζ(3) was irrational in a lecture. The announcement was met with criticism in part because Apéry was relatively unknown at the time. He also gave the lecture in French, made jokes throughout, and omitted several key explanations needed to follow the proof..

For example, there was an equation at the beginning of his lecture that no one knew but formed the core of his proof. When asked where this equation came from, Apéry is said to have answered "They grow in my garden," which was said to have caused many in the audience to stand up and leave the room.

However, someone in attendance had an electronic calculator (uncommon at the time) and with a short program, checked Apéry's equation and found it correct.

The equation in question is below, which was an unknown series representation of ζ(3) at the time:

With this expression, he was able to use a condition for irrationality that German mathematician Gustav Lejeune Dirichlet had derived in the 19th century. It states that a number χ is irrational if there are an infinite number of integers p and q, so that the following inequality is satisfied:

Here, c and δ denote constant values. Although the formula looks complicated, it basically means that χ can be approximated by fractions, but there is no fractional number that corresponds to χ exactly. Apéry succeeded in deriving this inequality for ζ(3), and thus the number is irrational.

In simpler terms, Apéry's proof constructed two sequences of integers:

a_n and b_n

such that

a_n / b_n

approximate ζ(3) far too well for a rational number.

This is the key philosophical idea. If a number is rational, there are limits to how accurately fractions can approximate it without eventually becoming exact. Apéry built approximations that violated those limits.

The machinery involved strange recursive sequences and combinatorial identities that seemed to come out of nowhere.

Even now, many mathematicians describe the proof as "magical".

To honor his work, the value of ζ(3) now bears his name and is known as Apéry's constant. This doesn't answer all of the questions associated with the number, however. We are still looking for a clear numerical value for ζ(3) that can be expressed with known constants, much in the same way ζ(2) is.

But regardless, Apéry's constant feels like an accident. It emerges from a simple series, has no known simple closed form, and required centuries to understand even partially. And yet it keeps appearing across math and physics like a recurring character in a story nobody fully understands.

Max Bill's Gelbes Feld

Cover art for this article by Barry Cipra in this month's Notices. Gelbes Feld = Yellow Field, and the original Max Bill piece is a magic square.

The math in the article is about pipifying odd magic squares.

But then my eye was caught by something else: The way the 4 appears in the lower right-hand corner is not the way it’s typically represented by pips on a die or domino. The “normal” way to display a 4 is with dots in the four corners, not at the four midpoints. That artistic choice bothered me. Until a possible explanation jumped out: Bill had arranged things so that there are exactly 5 pips across each row, down each column, and along the two main diagonals of his array. I’ll use the term “pipification” to describe such a representation of numbers in an 𝑛 × 𝑛 magic square using pips in an 𝑛 × 𝑛 grid within each of the 𝑛2 cells in the square, arranged so that there are the same number of pips in each row, column, and diagonal; see Figure 3.

Count the pips!

Made me make this...

Today it’s mostly famous for what it looks like on the inside.

It has an equestrian staircase though it’s so smooth it’s really just a gentle slope more than a staircase. It was build like that so our lazy bum king could ride his horse all the way to the top (king not in photo)

And naturally people have also driven cars up the tower

And held a bike race

For a while it was just sort of abandoned by the authorities and became a spiraling marketplace

But today it has been restored and become a tourist spot as well as a popular destination for school trips. And yes, you can still watch the cosmos at the top.